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Theorem 1st2nd 6182
Description: Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
Assertion
Ref Expression
1st2nd  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )

Proof of Theorem 1st2nd
StepHypRef Expression
1 df-rel 4712 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3188 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 458 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6175 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
53, 4syl 15 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   <.cop 3656    X. cxp 4703   Rel wrel 4710   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  2ndrn  6184  1st2ndbr  6185  elopabi  6201  cnvf1olem  6232  ordpinq  8583  addassnq  8598  mulassnq  8599  distrnq  8601  mulidnq  8603  recmulnq  8604  ltexnq  8615  fsumcnv  12252  cofulid  13780  cofurid  13781  idffth  13823  cofull  13824  cofth  13825  ressffth  13828  isnat2  13838  nat1st2nd  13841  homadmcd  13890  catciso  13955  prf1st  13994  prf2nd  13995  1st2ndprf  13996  curfuncf  14028  uncfcurf  14029  curf2ndf  14037  yonffthlem  14072  yoniso  14075  dprd2dlem2  15291  dprd2dlem1  15292  dprd2da  15293  2ndcctbss  17197  caubl  18749  rngoi  21063  drngoi  21090  nvop2  21180  nvvop  21181  nvop  21259  phop  21412  cvmliftlem1  23831  11st22nd  25148  heiborlem3  26640  isdrngo1  26690  iscrngo2  26726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139
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